Problem: Biologists breed a pair of plants where each plant has the color genes Gg. This means that any offspring these plants produce has a $25\%$ chance of inheriting the color genes gg, which makes the offspring an albino plant. Assume that offspring inherit genes independently from each other. Which of the following would find the probability of this pair producing exactly $1$ albino plant among $6$ offspring? Choose 1 answer: Choose 1 answer: (Choice A) A $(0.25)(0.75)^5$ (Choice B) B $(0.25)^5(0.75)$ (Choice C) C ${25 \choose 1}(0.25)(0.75)^5$ (Choice D) D ${6 \choose 1}(0.25)^5(0.75)$ (Choice E) E ${6 \choose 1}(0.25)(0.75)^5$
Explanation: Probability of $1$ albino offspring We want the probability that this pair produces $1$ success (albino plant) in $6$ trials (number of offspring), so we're going to need $5$ failures (non-albino plants) as well. The probability of each success is ${25\%}$ and the probability of each failure is $75\%}$. Since we were told to assume independence, we can multiply probabilities to find the probability of getting $1$ success followed by $5$ failures: $\begin{aligned} P(\text{SFFFFF})&=({0.25})(0.75})(0.75})(0.75})(0.75})(0.75}) \\\\ &=({0.25})(0.75})^5 \end{aligned}$ The binomial coefficient ${n \choose k}$ SFFFFF isn't the only arrangement that produces $1$ success in $6$ trials. For instance, FFFFFS would also produce the desired outcome. To count how many possible arrangements there are, we use the binomial coefficient ${n \choose k}$. It tells us the number of possible arrangements for $k$ successes in $n$ trials. In this problem, we want $k=1$ success (albino) in $n=6$ trials (offspring), so we should use the binomial coefficient ${6 \choose 1}$. [Tell me more about the binomial coefficient.] Putting it together Each arrangement has probability $(0.25)(0.75)^5$ so for our final answer we multiply this probability by the number of possible arrangements: ${6 \choose 1}(0.25)(0.75)^5$ The answer: ${6 \choose 1}(0.25)(0.75)^5$